Finite Group Databasebeta

← #21432129

Subgroups — #21432129

Order 7552.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273705C211yesmaximal, characteristic, complemented
2C2×#14273702C211yesmaximal, characteristic
3#16837337C212yesmaximal, complemented
4#16837333C211yesmaximal, characteristic, complemented
5#16837320C211yesmaximal, characteristic, complemented
6#16837320C212yesmaximal, complemented
7#16837319C211yesmaximal, characteristic, complemented
8#16837319C212yesmaximal, complemented
9#16837315C212yesmaximal, complemented
10#16837264C211yesmaximal, characteristic, complemented
11C59×#398C211yesmaximal, characteristic, complemented
12C22×Dic118C2211yescharacteristic
13C22×D236C2212yes—
14C22×D236C2211yescharacteristic
15C22×C4×D59C2211yescharacteristic
16#14273705—24no—
17#14273702—24no—
18C2×Dic59⋊C4C2211yescharacteristic
19C2×Dic59⋊C4C2211yescharacteristic
20C2×Dic59⋊C4C2212yes—
21C2×(C2×D59)⋊C4C2212yes—
22C2×(C2×D59)⋊C4C2211yescharacteristic
23C2×(C2×D59)⋊C4C2211yescharacteristic
24C2×(C2×D59)⋊C4C2212yes—
25C2×(C2×D59)⋊C4C2212yes—
26C2×(C2×D59)⋊C4C2212yes—
27C2×(C2×D59)⋊C4C2212yes—
28C2×(C2×D59)⋊C4C2212yes—
29C59⋊#95C2212yes—
30C59⋊#95C2211yescharacteristic
31C59⋊#95C2212yes—
32C2×C4⋊C4×C59C2211yescharacteristic
33C2×C4⋊C4×C59C2211yescharacteristic
34C2×C4⋊C4×C59C2212yes—
35#95×C59C2211yescharacteristic
36#95×C59C2212yes—
37C23×D59C2312yes—
38C23×D59C2311yescharacteristic
39C22×C4×C59C2312yes—
40C22×C4×C59C2311yescharacteristic
41C22×C4×C59C2312yes—
42C22×C4×C59C2311yescharacteristic
43C22×C4×C59C2311yescharacteristic
44C22×Dic59C2312yes—
45C22×Dic59C2311yescharacteristic
46C22×Dic59C2311yescharacteristic
47C22×Dic59C2311yescharacteristic
48C2×D236—24no—
49C2×D236—22no—
50C2×D236—42no—
51C2×D236—42no—
52C2×D236—42no—
53C2×C4×D59—42no—
54C2×C4×D59D414yes—
55C2×Dic118—22no—
56C2×Dic118—42no—
57C4⋊C4×C59—22no—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—42no—
60(C2×D59)⋊C4—22no—
61(C2×D59)⋊C4—22no—
62(C2×D59)⋊C4—24no—
63(C2×D59)⋊C4—42no—
64(C2×D59)⋊C4—42no—
65(C2×D59)⋊C4—42no—
66(C2×D59)⋊C4—42no—
67(C2×D59)⋊C4—42no—
68Dic59⋊C4—22no—
69Dic59⋊C4—22no—
70Dic59⋊C4—42no—
71C23×C59C2411yescharacteristic
72C22×D59—22no—
73C22×D59—21no—
74C22×D59—21no—
75C22×D59—22no—
76C22×D59—22no—
77C22×D59—22no—
78C22×D59—22no—
79C22×D59—22no—
80C22×D59—22no—
81C22×D59—22no—
82C22×D59—22no—
83C22×D59C2×D412yes—
84C2×C4×C59—22no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—22no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—22no—
96C2×C4×C59C2×D412yes—
97C2×Dic59—22no—
98C2×Dic59—21no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—22no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104C2×Dic59—22no—
105C2×Dic59—22no—
106C2×Dic59C2×D412yes—
107D236—44no—
108D236—48no—
109C4×D59—44no—
110Dic118—44no—
111C22×C59C22×D411yescharacteristic
112C22×C59H(5,𝔽2)11yescharacteristic
113C22×C59H(5,𝔽2)12yes—
114C22×C59H(5,𝔽2)11yescharacteristic
115C22×C592−1+412yes—
116C2×D59—22no—
117C2×D59—22no—
118C2×D59—22no—
119C2×D59—42no—
120C2×D59—42no—
121C2×D59—42no—
122C2×D59—42no—
123C2×D59—42no—
124C2×D59—42no—
125C2×D59—42no—
126C2×D59—42no—
127C2×D59—42no—
128C4×C59—22no—
129C4×C59—41no—
130C4×C59—42no—
131C4×C59—41no—
132C4×C59—42no—
133Dic59—22no—
134Dic59—41no—
135Dic59—41no—
136Dic59—42no—
137#2816—591nomaximal, Sylow
138C2×C59#53411yescharacteristic
139C2×C59#53511yescharacteristic
140C2×C59#53611yescharacteristic
141C2×C59#56212yes—
142C2×C59#56212yes—
143D59—42no—
144D59—82no—
145C2×C22⋊Q8—591no—
146C2×#121—591no—
147#399—592no—
148#398—591no—
149#398D5911yescharacteristic, complemented
150#394—591no—
151#394—592no—
152#394—591no—
153#394—592no—
154#393—592no—
155#381—591no—
156C59#281611yescharacteristic, complemented, Sylow
157C22×Q8—591no—
158C22×D4—592no—
159C22×D4—591no—
160C23×C4—591no—
161C22⋊Q8—1184no—
162#121—1184no—
163C2×C4⋊C4—591no—
164C2×C4⋊C4—591no—
165C2×C4⋊C4—592no—
166C2×C4⋊C4C2×D5911yescharacteristic
167C2×C4⋊C4C2×D5911yescharacteristic
168C2×C4⋊C4C2×D5912yes—
169C2×C22⋊C4—592no—
170C2×C22⋊C4—591no—
171C2×C22⋊C4—591no—
172C2×C22⋊C4—592no—
173C2×C22⋊C4—592no—
174C2×C22⋊C4—592no—
175C2×C22⋊C4—592no—
176C2×C22⋊C4—592no—
177#95—592no—
178#95—591no—
179#95—592no—
180#95C2×D5911yescharacteristic
181#95C2×D5912yes—
182C24—591no—
183C24—592no—
184C2×Q8—1182no—
185C2×Q8—2362no—
186C2×D4—1184no—
187C2×D4—1182no—
188C2×D4—2362no—
189C2×D4—2362no—
190C2×D4—2362no—
191C22×C4—591no—
192C22×C4—594no—
193C22×C4—592no—
194C22×C4—591no—
195C22×C4—591no—
196C22×C4—2362no—
197C22×C4C22×D5912yes—
198C22×C4C22×D5912yes—
199C22×C4C22×D5911yescharacteristic
200C22×C4C22×D5911yescharacteristic
201C22×C4C22×D5911yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—1182no—
206C4⋊C4—1182no—
207C4⋊C4—2362no—
208C22⋊C4—1184no—
209C22⋊C4—1182no—
210C22⋊C4—1182no—
211C22⋊C4—2362no—
212C22⋊C4—2362no—
213C22⋊C4—2362no—
214C22⋊C4—2362no—
215C22⋊C4—2362no—
216C23—592no—
217C23—1181no—
218C23—1182no—
219C23—1181no—
220C23—1182no—
221C23—1182no—
222C23—1182no—
223C23—1182no—
224C23—1182no—
225C23—1182no—
226C23—1182no—
227C23—1182no—
228C23C23×D5911yescharacteristic
229Q8—2364no—
230D4—2364no—
231D4—2368no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—592no—
245C2×C4—1182no—
246C2×C4—1181no—
247C2×C4—1181no—
248C2×C4—1181no—
249C2×C4—1182no—
250C2×C4—1182no—
251C2×C4—1182no—
252C2×C4—1181no—
253C2×C4—1182no—
254C2×C4—2364no—
255C2×C4D4×D5912yes—
256C22—1182no—
257C22—1182no—
258C22—1182no—
259C22—2362no—
260C22—2362no—
261C22—2362no—
262C22—2362no—
263C22—2362no—
264C22—2362no—
265C22—2362no—
266C22—2362no—
267C22—2362no—
268C22C2×D4×D5911yescharacteristic
269C22#1427377211yescharacteristic
270C22#1427377312yes—
271C22#1427377311yescharacteristic
272C22#1427377512yes—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—42no—
277C4—41no—
278C4—1182no—
279C4—2362no—
280C4—2361no—
281C4—2361no—
282C2—4722no—
283C2—2362no—
284C2#1683796311yescharacteristic
285C2#1683796811yescharacteristic
286C2#1683797711yescharacteristic
287C2#1683814812yes—
288C2#1683815012yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC59×#398C2
hypercenter#398D59
hypocenterC59#2816
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ #398

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#14273705C2semidirect
3. #16837337C2semidirect
4. #16837333C2semidirect
5. #16837320C2semidirect
6. #16837320C2semidirect
7. #16837319C2semidirect
8. #16837319C2semidirect
9. #16837315C2semidirect
10. #16837264C2semidirect
11. C59×#398C2semidirect
149. #398D59semidirect
156. C59#2816semidirect